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Fixed Point Theorems and Applications

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Metric Structures and Fixed Point Theory

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It is an indisputable argument that the formulation of metrics (by Fréchet in the early 1900s) opened a new subject in mathematics called non-linear analysis after the appearance of Banach’s fixed point theorem. Because the underlying space of this theorem is a metric space, the theory that developed following its publication is known as metric fixed point theory. It is well known that metric fixed point theory provides essential tools for solving problems arising in various branches of mathematics and other sciences such as split feasibility problems, variational inequality problems, non-linear optimization problems, equilibrium problems, selection and matching problems, and problems of proving the existence of solutions of integral and differential equations are closely related to fixed point theory. For this reason, many people over the past seventy years have tried to generalize the definition of metric space and corresponding fixed point theory. This trend still continues. A few questions lying at the heart of the theory remain open and there are many unanswered questions regarding the limits to which the theory may be extended.

Metric Structures and Fixed Point Theory provides an extensive understanding and the latest updates on the subject. The book not only shows diversified aspects of popular generalizations of metric spaces such as symmetric, b -metric, w -distance, G -metric, modular metric, probabilistic metric, fuzzy metric, graphical metric and corresponding fixed point theory but also motivates work on existing open problems on the subject. Each of the nine chapters—contributed by various authors—contains an Introduction section which summarizes the material needed to read the chapter independently of the others and contains the necessary background, several examples, and comprehensive literature to comprehend the concepts presented therein. This is helpful for those who want to pursue their research career in metric fixed point theory and its related areas.

• Explores the latest research and developments in fixed point theory on the most popular generalizations of metric spaces
• Description of various generalizations of metric spaces
• Very new topics on fixed point theory in graphical and modular metric spaces
• Enriched with examples and open problems

This book serves as a reference for scientific investigators who need to analyze a simple and direct presentation of the fundamentals of the theory of metric fixed points. It may also be used as a text book for postgraduate and research students who are trying to derive future research scope in this area.

TABLE OF CONTENTS

Chapter 1 | 30  pages, symmetric spaces and fixed point theory, chapter chapter 2 | 34  pages, fixed point theory in b-metric spaces, chapter chapter 3 | 36  pages, basics of w-distance and its use in various types of results, chapter chapter 4 | 46  pages, g-metric spaces: from the perspective of f-contractions and best proximity points, chapter chapter 5 | 50  pages, fixed point theory in probabilistic metric spaces, chapter chapter 6 | 46  pages, fixed point theory for fuzzy contractive mappings, chapter chapter 7 | 22  pages, set-valued maps and inclusion problems inmodular metric spaces, chapter chapter 8 | 16  pages, graphical metric spaces and fixed point theorems, chapter chapter 9 | 16  pages, fixed point theory in partial metric spaces.

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New Challenges and Trends in Fixed Point Theory and Its Applications

This thematic series is devoted to publishing the latest and most significant research on Fixed Point Theory including its wide range of applications. Its goals are to stimulate further research and to highlight and emphasize the most recent advances in the field as well as to promote, encourage, and bring together researchers in Fixed point Theory and Applications including those interested in potential applications in Science and Engineering.

Edited by: Yeol Je Cho, Manuel De la Sen, Abdul Latif and Reza Saadati

Fixed point theorems for multivalued mappings in ordered Banach spaces with application to integral inclusions

This paper provides new common fixed point theorems for pairs of multivalued and single-valued mappings operating between ordered Banach spaces. Our results lead to new existence theorems for a system of integ...

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Existence of solution and algorithms for a class of bilevel variational inequalities with hierarchical nesting structure

In this paper we consider a class of bilevel variational inequalities with hierarchical nesting structure. We first of all get the existence of a solution for this problem by using the Himmelberg fixed point t...

Fixed point and periodic point results for α -type F -contractions in modular metric spaces

Motivated by Gopal et al. (Acta Math. Sci. 36B(3):1-14, 2016 ). We introduce the notion of α -type F -contraction in the setting of modular metric spaces which is independent from one given in (Hussain et al. in Fix...

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Variational inequality formulation of circular cone eigenvalue complementarity problems.

In this paper, we study the circular cone eigenvalue complementarity problem (CCEiCP) by variational inequality technique, prove the existence of a solution to CCEiCP, and investigate different nonlinear progr...

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In this paper, we first propose a weak convergence algorithm, called the linesearch algorithm, for solving a split equilibrium problem and nonexpansive mapping (SEPNM) in real Hilbert spaces, in which the firs...

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A coupled fixed point theorem and application to fractional hybrid differential problems, common fixed point results of a pair of generalized \((\psi,\varphi )\) -contraction mappings in modular spaces.

In this paper, we establish the existence of a common fixed point of almost generalized contractions on modular spaces. As an application, we present some fixed and common fixed point results for such mappings...

The Banach contraction principle in \(C^{*}\) -algebra-valued b -metric spaces with application

On best proximity points of upper semicontinuous multivalued mappings, unification of several distance functions and a common fixed point result, iterative algorithms for infinite accretive mappings and applications to p -laplacian-like differential systems.

Some new iterative algorithms with errors for approximating common zero point of an infinite family of m -accretive mappings in a real Banach space are presented. A path convergence theorem and some new weak and s...

Iterative algorithm for a family of split equilibrium problems and fixed point problems in Hilbert spaces with applications

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Extensions of almost- F and F -Suzuki contractions with graph and some applications to fractional calculus

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A comparative study on the convergence rate of some iteration methods involving contractive mappings

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Coupled fixed point theorems for single-valued operators in b -metric spaces

The aim of this paper to present fixed point results for single-valued operators in b -metric spaces. The case of scalar metric and the case of vector-valued metric approaches are considered. As an application, a ...

About periodicity of impulsive evolution equations through fixed point theory

By processing the problem through fixed point theory and propagator theory, we investigate the periodicity of solutions to a class of impulsive evolution equations in Hilbert spaces and establish some existenc...

Fixed point theorems in locally convex spaces and a nonlinear integral equation of mixed type

In this paper, we provide a new approach for discussing the solvability of a class of operator equations by establishing fixed point theorems in locally convex spaces. Our results are obtained extend some Kras...

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Strong convergence theorems for asymptotically nonexpansive nonself-mappings with applications

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Existence of best proximity points for controlled proximal contraction

In this paper, we investigate the sufficient condition for the existence of best proximity points for non-self-multivalued mappings. Additionally, we discuss the stability theorem for such mappings. Our result...

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An intermixed algorithm for two strict pseudo-contractions in Hilbert spaces have been presented. It is shown that the suggested algorithms converge strongly to the fixed points of two strict pseudo-contractio...

Self-adaptive algorithms for proximal split feasibility problems and strong convergence analysis

The purpose of the paper is to study the proximal split feasibility problems. For solving the problems, we present new self-adaptive algorithms with the regularization technique. By using these algorithms, we ...

General iterative methods for monotone mappings and pseudocontractive mappings related to optimization problems

In this paper, we introduce two general iterative methods for a certain optimization problem of which the constrained set is the common set of the solution set of the variational inequality problem for a conti...

New result on fixed point theorems for φ -contractions in Menger spaces

Very recently, Fang (Fuzzy Sets Syst. 267:86-99, 2015) gave some fixed point theorems for probabilistic φ -contractions in Menger spaces. Fang’s results improve the one of Jachymski (Nonlinear Anal. 73:2199-2203, ...

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Common fixed point theorems for Geraghty’s type contraction mappings using the monotone property with two metrics

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Coincidence point and fixed point theorems for a new type of G -contraction multivalued mappings on a metric space endowed with a graph

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Title: common fixed point theorems for a commutative family of nonexpansive mappings in complete random normed modules.

Abstract: In this paper, we first introduce and study the notion of random Chebyshev centers. Further, based on the recently developed theory of stable sets, we introduce the notion of random complete normal structure so that we can prove the two deeper theorems: one of which states that random complete normal structure is equivalent to random normal structure for an $L^0$-convexly compact set in a complete random normed module; the other of which states that if $G$ is an $L^0$-convexly compact subset with random normal structure of a complete random normed module, then every commutative family of nonexpansive mappings from $G$ to $G$ has a common fixed point. We also consider the fixed point problems for isometric mappings in complete random normed modules. Finally, as applications of the fixed point theorems established in random normed modules, when the measurable selection theorems fail to work, we can still prove that a family of strong random nonexpansive operators from $(\Omega,\mathcal{F},P)\times C$ to $C$ has a common random fixed point, where $(\Omega,\mathcal{F},P)$ is a probability space and $C$ is a weakly compact convex subset with normal structure of a Banach space.

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Advances in Metric Fixed Point Theory and Applications

• © 2021
• Yeol Je Cho 0 ,
• Mohamed Jleli 1 ,
• Mohammad Mursaleen 2 ,
• Bessem Samet 3 ,
• Calogero Vetro 4

Department of Mathematics Education, Gyeongsang National University, Jinju, Korea (Republic of)

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Department of Mathematics, King Saud University, Riyadh, Saudi Arabia

Department of mathematics, aligarh muslim university, aligarh, india, department of mathematics and computer sciences, university of palermo, palermo, italy.

• Presents a detailed treatment of fixed point theory and its applications
• Includes rigorous mathematical proofs, recent developments, and approaches associated with fixed point theory
• Appeals to graduate students, teachers, and researchers alike

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About this book

This book collects papers on major topics in fixed point theory and its applications. Each chapter is accompanied by basic notions, mathematical preliminaries and proofs of the main results. The book discusses common fixed point theory, convergence theorems, split variational inclusion problems and fixed point problems for asymptotically nonexpansive semigroups; fixed point property and almost fixed point property in digital spaces, nonexpansive semigroups over CAT(κ) spaces, measures of noncompactness, integral equations, the study of fixed points that are zeros of a given function, best proximity point theory, monotone mappings in modular function spaces, fuzzy contractive mappings, ordered hyperbolic metric spaces, generalized contractions in b-metric spaces, multi-tupled fixed points, functional equations in dynamic programming and Picard operators.

This book addresses the mathematical community working with methods and tools of nonlinear analysis. It also serves as a reference, source for examples and new approaches associated with fixed point theory and its applications for a wide audience including graduate students and researchers.

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Genericity and porosity in fixed point theory: a survey of recent results

Introduction to Metric Fixed Point Theory

Survey on Metric Fixed Point Theory and Applications

• fixed point theory
• common fixed point
• metric fixed point theory
• coupled fixed point
• generalized contraction
• best proximity
• Contor’s theorem
• simulation functions
• Picard operators
• polynomial approximations

Table of contents (20 chapters)

Front matter, the relevance of a metric condition on a pair of operators in common fixed point theory.

• A. Petruşel, I. A. Rus

Some Convergence Results of the \(K^{*}\) Iteration Process in CAT(0) Spaces

• Aynur Şahin, Metin Başarır

Split Variational Inclusion Problem and Fixed Point Problem for Asymptotically Nonexpansive Semigroup with Application to Optimization Problem

• Shih-sen Chang, Liangcai Zhao, Zhaoli Ma

Convergence Theorems and Convergence Rates for the General Inertial Krasnosel’skiǐ–Mann Algorithm

• Qiao-Li Dong, Shang-Hong Ke, Yeol Je Cho, Themistocles M. Rassias

Digital Space-Type Fixed Point Theory and Its Applications

• Sang-Eon Han

Existence and Approximations for Order-Preserving Nonexpansive Semigroups over \(\mathrm{CAT}(\kappa )\) Spaces

• Parin Chaipunya

A Solution of the System of Integral Equations in Product Spaces via Concept of Measures of Noncompactness

• Hemant Kumar Nashine, Reza Arab, Rabha W. Ibrahim

Fixed Points That Are Zeros of a Given Function

• Francesca Vetro

A Survey on Best Proximity Point Theory in Reflexive and Busemann Convex Spaces

• Moosa Gabeleh

On Monotone Mappings in Modular Function Spaces

• M. R. Alfuraidan, M. A. Khamsi, W. M. Kozlowski

Contributions to Fixed Point Theory of Fuzzy Contractive Mappings

• Dhananjay Gopal

Common Fixed Point Theorems for Four Maps

• Muhammad Nazam, Choonkil Park, Muhammad Arshad

Measure of Noncompactness in Banach Algebra and Its Application on Integral Equations of Two Variables

• Anupam Das, Bipan Hazarika

Generalization of Darbo-Type Fixed Point Theorem and Applications to Integral Equations

• Hemant Kumar Nashine, Rabha W. Ibrahim, Reza Arab, M. Rabbani

Approximating Fixed Points of Suzuki \((\alpha ,\beta )\) -Nonexpansive Mappings in Ordered Hyperbolic Metric Spaces

• Juan Martínez-Moreno, Kenyi Calderón, Poom Kumam, Edixon Rojas

Generalized JS -Contractions in b -Metric Spaces with Application to Urysohn Integral Equations

• Hemant Kumar Nashine, Zoran Kadelburg

Unified Multi-tupled Fixed Point Theorems Involving Monotone Property in Ordered Metric Spaces

• Mohammad Imdad, Aftab Alam, Javid Ali, Stojan Radenović

Convergence Analysis of Solution Sets for Minty Vector Quasivariational Inequality Problems in Banach Spaces

• Nguyen Van Hung, Dinh Huy Hoang, Vo Minh Tam, Yeol Je Cho

Common Solutions for a System of Functional Equations in Dynamic Programming Passing Through the JCLR -Property in \(S_b\) -Metric Spaces

• Oratai Yamaod, Wutiphol Sintunavarat, Yeol Je Cho

Editors and Affiliations

Yeol Je Cho

Mohamed Jleli, Bessem Samet

Mohammad Mursaleen

Calogero Vetro

About the editors

YEOL JE CHO is Emeritus Professor at the Department of Mathematics Education, Gyeongsang National University, Jinju, Korea, and Distinguished Professor at the School of Mathematical Sciences, the University of Electronic Science and Technology of China, Chengdu, Sichuan, China. In 1984, he received his Ph.D. in Mathematics from Pusan National University, Pusan, Korea. He is a fellow of the Korean Academy of Science and Technology, Seoul, Korea, since 2006, and a member of several mathematical societies. He has organized international conferences on nonlinear functional analysis and applications, fixed point theory and applications and workshops and symposiums on nonlinear analysis and applications. He has published over 400 papers, 20 monographs and 12 books with renowned publishers from around the world. His research areas are nonlinear analysis and applications, especially fixed point theory and applications, some kinds of nonlinear problems, that is, equilibrium problems, variational inequality problems, saddle point problems, optimization problems, inequality theory and applications, stability of functional equations and applications. He has delivered several invited talks at international conferences on nonlinear analysis and applications and is on the editorial boards of 10 international journals of mathematics.

MOHAMED JLELI is Full Professor of Mathematics at King Saud University, Saudi Arabia. He received his Ph.D. in Pure Mathematics with the thesis entitled “Constant Mean Curvature Hypersurfaces” from the Faculty of Sciences of Paris VI, France, in 2004. His research interests include surfaces and hypersurfaces in space forms, mean curvature, nonlinear partial differential equations, nonlinear fractional calculus and nonlinear analysis, on which he has published his research articles in international journals of repute. He is on the editorial board member of several international journals of mathematics. 

MOHAMMAD MURSALEEN is Professor at the Department of Mathematics, Aligarh Muslim University (AMU), India. He is currently Principal Investigator for a SERB Core Research Grant at the Department of Mathematics, AMU, India. He is also Visiting Professor at China Medical University, Taiwan, since January 2019. He has served as Lecturer to Full Professor at AMU since 1982 and as Chair of the Department of Mathematics from 2015 to 2018. He has published more than 350 research papers in the field of summability, sequence spaces, approximation theory, fixed point theory and measures of noncompactness and has authored/co-edited 9 books. Besides several master’s students, he has guided 21 Ph.D. students. He served as a reviewer for various international scientific journals and is on the editorial boards of many international scientific journals. He is on the list of Highly Cited Researchers for the year 2019 of Thomson Reuters (Web of Science). 

BESSEM SAMET is Full Professor of Applied Mathematics at King Saud University, Saudi Arabia. He received his Ph.D. in Applied Mathematics with the thesis entitled “Topological Derivative Method for Maxwell Equations and its Applications” from Paul Sabatier University, France, in 2004. His areas of research include different branches of nonlinear analysis, including fixed point theory, partial differential equations, fractional calculus and more. He has authored/co-authored over 100 published research papers in ISI journals. He was on the list of Thomson Reuters Highly Cited Researchers for the years 2015 to 2017. 

CALOGERO VETRO is Assistant Professor of Mathematical Analysis at the University of Palermo, Italy, since 2005. He is also affiliated with the Department of Mathematics and Computer Science of the university. He received his Ph.D. in Engineering of Automation and Control Systems in 2004 and the Laurea Degree in Mechanical Engineering in 2000. He has taught courses in mathematical analysis, numerical analysis, numerical calculus, geomathematics, computational mathematics, operational research and optimization. He is a member of Doctoral Collegium at the University of Palermo and acts as a referee for several scientific journals of pure and applied mathematics. He is also on the editorial boards of renowned scientific journals and a guest editor of special issues on fixed point theory and partial differential equations. His research interests include approximation, fixed point theory, functional analysis, mathematical programming, operator theory and partial differential equations. He has authored/co-authored over 150 published papers and was on the Thomson Reuters Highly Cited Researchers List from 2015 to 2017.

Bibliographic Information

Book Title : Advances in Metric Fixed Point Theory and Applications

Editors : Yeol Je Cho, Mohamed Jleli, Mohammad Mursaleen, Bessem Samet, Calogero Vetro

DOI : https://doi.org/10.1007/978-981-33-6647-3

Publisher : Springer Singapore

eBook Packages : Mathematics and Statistics , Mathematics and Statistics (R0)

Copyright Information : The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2021

Hardcover ISBN : 978-981-33-6646-6 Published: 04 May 2021

Softcover ISBN : 978-981-33-6649-7 Published: 05 May 2022

eBook ISBN : 978-981-33-6647-3 Published: 04 May 2021

Edition Number : 1

Number of Pages : XVII, 503

Number of Illustrations : 5 b/w illustrations, 1 illustrations in colour

Topics : Functional Analysis , Topology , Operator Theory

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Fixed point theory : Banach, Brouwer and Schauder theorems

• Masters Thesis
• Shamash, Ellis Raymond
• Breen, Stephen
• Biriuk, George
• Mathematics
• California State University, Northridge
• Dissertations, Academic -- CSUN -- Mathematics.
• 2016-06-14T17:03:50Z
• http://hdl.handle.net/10211.3/172493
• by Ellis Raymond Shamash
• California State University, Northridge. Department of Mathematics.
• Includes bibliographical references (leaves 64-66)

Items in ScholarWorks are protected by copyright, with all rights reserved, unless otherwise indicated.

• DOI: 10.1090/S0273-0979-1983-15153-4
• Corpus ID: 33715682

Fixed point theory and nonlinear problems

• Published 1 July 1983
• Mathematics
• Bulletin of the American Mathematical Society

275 Citations

History of the brouwer fixed point theorem, introduction to: topological degree and fixed point theories in differential and difference equations, variational principles and critical point theory, partial differential equations in the 20 th century *, a topological degree of type (s) for hammerstein operators, variational inequalities,bifurcation and applications, the bolzano mean-value theorem and partial differential equations, on the periodic solutions of discontinuous piecewise differential systems, the degree theory for set-valued compact perturbation of monotone-type mappings with an application.

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Fixed Point Theorem and Related Nonlinear Analysis by the Best Approximation Method in p-Vector Spaces

31 references, on the uniqueness of the topological degree, the topological degree for noncompact nonlinear mappings in banach spaces, topology and nonlinear boundary value problems, nonlinear functional analysis, degree of mapping for nonlinear mappings of monotone type., the leray-schauder index and the fixed point theory for arbitrary anrs, some historical remarks concerning degree theory, topologie et équations fonctionnelles.

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On the uniqueness of the topological degree fork-set-contractions

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Applications in Fixed Point Theory

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Banach's contraction principle is probably one of the most important theorems in fixed point theory. It has been used to develop much of the rest of fixed point theory. Another key result in the field is a theorem due to Browder, Göhde, and Kirk involving Hilbert spaces and nonexpansive mappings. Several applications of Banach's contraction principle are made. Some of these applications involve obtaining new metrics on a space, forcing a continuous map to have a fixed point, and using conditions on the boundary of a closed ball in a Banach space to obtain a fixed point. Finally, a development … continued below

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Farmer, Matthew Ray December 2005.

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• Farmer, Matthew Ray
• Bator, Elizabeth M.

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• Lewis, Paul
• Jackson, Stephen C.
• University of North Texas Place of Publication: Denton, Texas

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• Name: Master of Arts
• Level: Master's
• Discipline: Mathematics
• Department: Department of Mathematics
• Grantor: University of North Texas

Banach's contraction principle is probably one of the most important theorems in fixed point theory. It has been used to develop much of the rest of fixed point theory. Another key result in the field is a theorem due to Browder, Göhde, and Kirk involving Hilbert spaces and nonexpansive mappings. Several applications of Banach's contraction principle are made. Some of these applications involve obtaining new metrics on a space, forcing a continuous map to have a fixed point, and using conditions on the boundary of a closed ball in a Banach space to obtain a fixed point. Finally, a development of the theorem due to Browder et al. is given with Hilbert spaces replaced by uniformly convex Banach spaces.

• Banach spaces
• contraction maps
• fixed points
• metric space
• non-expansive maps
• uniformly convex Banach spaces

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• Fixed point theory.
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• Archival Resource Key : ark:/67531/metadc4971

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Farmer, Matthew Ray. Applications in Fixed Point Theory , thesis , December 2005; Denton, Texas . ( https://digital.library.unt.edu/ark:/67531/metadc4971/ : accessed August 24, 2024 ), University of North Texas Libraries, UNT Digital Library, https://digital.library.unt.edu ; .